What is the area under the curve y = x^4 from x = 0 to x = 1?

What is the area under the curve y = x^4 from x = 0 to x = 1?

Correct Answer: 1/5

Step 1: Understand that we want to find the area under the curve of the function y = x^4 between x = 0 and x = 1.
Step 2: To find the area, we need to use integration. We will set up the integral as ∫(from 0 to 1) x^4 dx.
Step 3: Calculate the integral of x^4. The integral of x^n is (x^(n+1))/(n+1). Here, n = 4, so we get (x^(4+1))/(4+1) = x^5/5.
Step 4: Now we need to evaluate this integral from 0 to 1. This means we will calculate [x^5/5] from 0 to 1.
Step 5: First, plug in the upper limit (1): (1^5)/5 = 1/5.
Step 6: Next, plug in the lower limit (0): (0^5)/5 = 0.
Step 7: Now, subtract the lower limit result from the upper limit result: (1/5) - (0) = 1/5.
Step 8: Therefore, the area under the curve y = x^4 from x = 0 to x = 1 is 1/5.

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